Notation Index. gcd(a, b) (greatest common divisor) NT-16
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1 Notation Index (for all) B A (all functions) B A = B A (all functions) SF-18 (n) k (falling factorial) SF-9 a R b (binary relation) C(n,k) = n! k! (n k)! (binomial coefficient) SF-9 n! (n factorial) SF-9 ( n ) k = n! k!(n k)! (binomial coefficient) SF-9 B n (Bell number) SF-11 χ (characteristic function) SF-10 (difference operator) IS-6 k n (k divides n; n/k Z) NT-2 x y (equivalence relation) EO-1! (for exactly one) (for some) Function χ (characteristic) SF-10 C(n,k) = ( n k) (binomial coefficient) SF-9 PER(A) = S(A) (permutations) SF-18 Coimage(f) SF-23 Image(f) SF-23 Function (particular) x (greatest integer) NT-9 x (ceiling) NT-9 gcd(a, b) (greatest common divisor) NT-16 φ(n) (Euler φ) NT-19 lcm(a,b) (least common multiple) NT-16 Function notation B A (all functions), SF-17, SF-18 f : A B (a function) BF-1, SF-15 f 1 (inverse, 1/f) SF-18 g f (composition) SF-20 gcd(a, b) (greatest common divisor) NT-16 lcm(a,b) (least common multiple) NT-16! (for exactly one) (for some) (for all) Logic notation (for some) Lo-13 (for all) Lo-12 (not) Lo-2 (and) Lo-2 (if and only if) Lo-6 (or) Lo-2 (if... then) Lo-5 x%d (x mod d) NT-7 N (Natural numbers) Lo-13, NT-1 n = {1,2,...,n} x C y (covering relation) EO-28 x y (order relation) EO-12 P (Prime numbers) Lo-13 P(A) (set of subsets of A) P k (A) (set of k-subsets of A) SF-9 SF-9 PER(A) = S(A) (permutations) SF-18 Q (Rational numbers) NT-1 R (Real numbers) Lo-13, NT-1 R(z) (real part of z) IS-24 Set notation {x : } (set description) SF-2 {x } (set description) SF-2 (empty set) SF-2 A (complement) SF-2 and (in and not in) SF-1 k A (k-fold product) SF-2 A (complement) SF-2 A B (difference) SF-2 A B (intersection) SF-2 A B (union) SF-2 A B (symmetric difference) SF-2 Index-1
2 A \ B (difference) SF-2 A B (subset) SF-1 A B (Cartesian product) SF-2 A c (complement) SF-2 P(A) (set of subsets of A) SF-9 P k (A) (set of k-subsets of A) SF-9 A (cardinality) SF-1 Sets of numbers N (Natural numbers) Lo-13, NT-1 N + (Positive integers) NT-1 N + 2 ({n Z n 2}) NT-1 P (Prime numbers) Lo-13, NT-2 Q (Rationals) NT-1 R (Real numbers) Lo-13, NT-1 Z (Integers) Lo-13, NT-1 n = {1,2,...,n} dz + k (residue class) NT-6 S(n,k) (Stirling number) SF-24 Z (Integers) Lo-13, NT-1 Index-2
3 Subject Index Index Absolute convergence IS-26 Absorption rule Adder full BF-19 half BF-18 BF-6, Lo-3, SF-3 Algebraic number theory NT-3 Algebraic rules for Boolean functions BF-6 predicate logic Lo-19 sequences IS-16 sets SF-2 statement forms Lo-3 Algorithm Euclidean NT-18 Alternating series IS-24 Dirichlet s Theorem IS-24 harmonic IS-23 And form BF-6 And operator (= ) BF-3 Antisymmetric relation EO-13 Arithmetic binary BF-12 computer BF-11 modular NT-6 two s complement BF-16 Associative law functional composition SF-20 Associative rule BF-6, Lo-3, SF-3 Base case (induction) IS-1 Base-b number BF-10 base change BF-10 binary (= base-2) BF-11 hexadecimal (= base-16) BF-11 octal (= base-8) BF-11 Bell number SF-11 Biconditional (= if and only if) Lo-6 Bijective function SF-18 Binary number BF-11 addition circuit BF-18 arithmetic BF-12 overflow BF-17 register size BF-14 two s complement BF-16 Binary operator BF-3 Binary relation EO-3 direct product of EO-18 Binomial coefficient Pascal s triangle SF-10 recursion SF-10 Binomial coefficient: C(n, k) = ( n k) SF-9 Block of a partition SF-11 Boolean operator, see also operator product (= ) EO-27 sum (= ) EO-27 Boolean function BF-1 number of BF-2 tabular form BF-1 Bound rule BF-6, Lo-3 Bounded sequence IS-16 monotone converge IS-17 Bucket sort EO-22 Cardinality of a set SF-1 Cartesian product of sets SF-2 Ceiling function (= least integer) NT-9 Chain (= linear order) EO-14 length of EO-29 Characteristic function SF-10 Ciphertext NT-13 Circuit for addition BF-18 Codomain of a function BF-1, SF-15 Index-3
4 Coimage of a function SF-23 set partition SF-23 Commutative rule BF-6, Lo-3, SF-3 Comparable elements EO-14 Comparison sort EO-22 Complement of a set SF-2 Composite number Lo-13, NT-2 Composition of functions SF-20, SF-20 associative law SF-20 Computer arithmetic addition circuit BF-18 negative number BF-16 overflow BF-14, BF-17 register size BF-14 two s complement BF-16 Conditional (= if... then) Lo-5 Conditional convergence IS-27 Conjecture Goldbach s Lo-13 Twin Prime Lo-16 Conjunctive normal form BF-6 Contradiction Lo-2 Contrapositive Lo-6 Convergence only tails matter IS-13 sequence IS-13 sequence alternate form IS-14 sequence bounded monotone IS-17 sequence to infinity IS-19 series IS-20 series Abel s Theorem IS-28 series absolute IS-26 series conditional IS-27 series general harmonic IS-25 series integral test IS-24 Converse Lo-6 Coordinate order (= direct product) EO-17 Countable set NT-5 Covering relation EO-28 Cryptography NT-13, SF-19 Diffie-Hellman protocol NT-22 PGP NT-20 public key NT-21 RSA protocol NT-23 symmetric encryption NT-20 trapdoor function NT-21 Cycle form of a permutation SF-22 Decreasing sequence IS-17 DeMorgan s rule DES (= Data Encryption Standard) SF-19 Diagonal argument NT-6 Diagram, Hasse BF-6, Lo-3, SF-3 EO-28 Dictionary order (= lex order) SF-8 Difference of sets SF-2 symmetric SF-2 Difference operator IS-6 Diffie-Hellman protocol NT-22 Digit symbol of index i BF-10 Direct product of binary relations EO-18 Direct product of posets EO-17 Directed graph diagrams EO-26 Discrete logarithm NT-21 Diffie-Hellman and NT-22 Disjunctive normal form BF-5 Distributive rule BF-6, Lo-3, SF-3 Divergence only tails matter IS-13 sequence IS-13 sequence to infinity IS-19 series IS-21 series to infinity IS-21 Divisible by: k n NT-2 Domain of a function BF-1, SF-15 Domino coverings EO-24 Double implication (= if and only if) Lo-6 Index-4
5 Double negation rule BF-6, Lo-3, SF-3 Element in poset greatest EO-29 least EO-29 maximal EO-30 minimal EO-30 Element method of proof SF-4 Elements (of a poset) comparable EO-14 incomparable EO-14 Empty set SF-2 Encryption SF-19 English to logic for all Lo-12 for some Lo-13 if and only if Lo-7 method for implication necessary Lo-7 neither BF-8 only if Lo-7 requires Lo-8 sufficient Lo-7 there exists Lo-13 unless Lo-8 Envelope game SF-17 Equivalence class EO-1 Equivalence relation EO-1 Espionage NT-15 Euclidean algorithm NT-18 Euler φ function NT-19 RSA protocol and NT-23 Even integer NT-1 Lo-8 Exclusive or operator (= ) BF-3 Existential quantifier ( ) Lo-13 Exponential, rate of growth of IS-18 Extension, linear EO-30 Factorial falling SF-9 Factoring RSA and NT-23 uniqueness of NT-3 Falling factorial SF-9 Fermat number Lo-16 Fermat s Last Theorem Lo-18, NT-3 Floor function (= greatest integer) NT-9 For all (logic: ) Lo-12 For some (logic: ) Lo-13 Full adder BF-19 Function BF-1, SF-15 bijective SF-18 binomial coefficient: ( n k) = C(n, k) SF-9 Boolean BF-1 Boolean, number of BF-2 ceiling (= least integer: x ) NT-9 characteristic: χ SF-10 codomain (= range) of BF-1, SF-15 coimage and set partition SF-23 coimage of SF-23 composition of SF-20, SF-20 domain of BF-1, SF-15 Euler φ NT-19 Euler φ and RSA protocol NT-23 floor (= greatest integer: x ) NT-9 greatest common divisor (= gcd) NT-16 greatest integer NT-9 hash SF-19 image of SF-15, SF-23 injective SF-18 inverse SF-18, SF-23 least common multiple (= lcm) NT-16 least integer NT-9 number of SF-18 number of = ( n k) S(m,k) k! SF-25 one-line notation for, Index-5
6 one-to-one (= injection) SF-18 one-way (= trapdoor) NT-21 onto (= surjection) SF-18 permutation SF-18 range (= codomain) of BF-1, SF-15 surjective SF-18 trapdoor NT-21 two-line notation for SF-20, SF-20 Functional relation Gate BF-18 Geometric series IS-22 Goldbach s conjecture Lo-13 Graph diagrams, directed EO-26 Greatest common divisor (= gcd) NT-16 Euclidean algorithm NT-18 Greatest element in poset EO-29 Greatest integer function NT-9 Half adder BF-18 Harmonic series IS-22 alternating IS-23 general IS-25 Hashing SF-19 Hasse diagram EO-28 Hexadecimal number BF-11 Idempotent rule BF-6, Lo-3, SF-3 If... then Lo-5 If and only if (logic) Lo-7 Image of a function SF-15, SF-23 Implication Lo-5 Incidence matrix EO-14 Incomparable elements EO-14 Incomparable subsets EO-14 Increasing sequence IS-17 Induction terminology IS-1 Inductive step Infinite sequence see Sequence Infinite series see Series IS-1 Injective function SF-18 Integral test for series IS-24 Intersection of sets SF-2 Inverse Lo-6 Inverse function SF-18 Inverse relation Irrationality of square root NT-4 Key (cryptography) NT-13 Diffie-Hellman NT-22 RSA and public NT-23 trapdoor function and NT-21 Lattice of subsets EO-13 Least common multiple (= lcm) NT-16 Least element in poset EO-29 Least integer function NT-9 Length-first lex order EO-21 Lexicographic bucket sort EO-22 Lexicographic order (= lex order) SF-7, EO-19 length-first (= short) EO-21 Limit of a sequence IS-13 sum of infinite series IS-20 Linear extension EO-30 Linear order SF-1, EO-14 List (= ordered set) SF-1 Logarithm discrete and Diffie- Hellman NT-22 Logarithm, rate of growth of IS-18 Logic predicate Lo-12 propositional BF-4, Lo-1 Index-6
7 Logic gate BF-18 Matrix, incidence EO-14 Maximal element in poset EO-30 Mersenne number Lo-17 Minimal element in poset EO-30 Mod as binary operator NT-7 Mod as equivalence relation NT-7 Modular arithmetic NT-6 Monotone sequence IS-17 Monotone subsequences EO-8 Necessary (logic) Lo-7 Negation rule BF-6, Lo-3 Normal form conjunctive BF-6 disjunctive BF-5 Not operator (= ) BF-3 Number base-b BF-10 Bell number: B n SF-11 binomial coefficient: ( n k) = C(n, k) SF-9 composite Lo-13, NT-2 Fermat: F n Lo-16 integer Z NT-1 integer: Z Lo-13 irrational: R Q NT-1 Mersenne: M p Lo-17 natural N NT-1 natural: N Lo-13 perfect Lo-17 prime Lo-13 prime: P Lo-13, NT-2 rational: Q NT-1 real: R Lo-13, NT-1 square root is irrational NT-4 Stirling: S(n, k) SF-24 unique prime factorization of NT-3 Number theory algebraic NT-3 elementary Lo-13 nonunique factorization NT-3 Octal number BF-11 Odd integer NT-1 One-line notation, One-to-one function (= injection) SF-18 One-way (= trapdoor) function NT-21 Only if (logic) Lo-7 Onto function (= surjection) SF-18 Operator and (= ) BF-3 binary BF-3 exclusive or (= ) BF-3 not (= ) BF-3 or (= ) BF-3 unary BF-3 Or form BF-5 Or operator (= ) BF-3 Order coordinate (= direct product) EO-17 dictionary (= lex) SF-8 lex (= lexicographic) SF-7 lexicographic EO-19 linear SF-1 relation SF-8 Order relation EO-12 Ordered set SF-1 Overflow BF-14, BF-17 Partially ordered set see poset Partition of a set SF-11 block of SF-11 function coimage and SF-23 number of SF-11, SF-24 refinement of SF-11 Pascal s triangle SF-10 Index-7
8 Perfect number Lo-17 Perfect square NT-4 Permutation SF-18 cycle SF-22 cycle form SF-22 cycle length SF-22 PGP (= Pretty Good Privacy) NT-20, SF-19 Pigeonhole principle extended EO-7 Plaintext NT-13 EO-5 Polynomial, rate of growth of IS-18 Poset EO-13 comparable elements EO-14 coordinate (= direct product) order EO-17 covering relation EO-28 direct product of EO-17 divisibility EO-14, EO-19 greatest element EO-29 incomparable elements EO-14 isomorphic EO-18 least element EO-29 lex order EO-19 linear (= total) order EO-14 maximal element EO-30 minimal element EO-30 restriction of (= subposet) EO-17 subset lattice EO-13, EO-17 Power set SF-9, EO-13 Powers sum of IS-5 Predicate logic algebraic rules Lo-19 predicate Lo-12 quantifier Lo-12 truth set Lo-12 Prime factorization NT-3, IS-2 uniqueness of NT-3 Prime number Lo-13, NT-2 how common? IS-28 infinitely many NT-4 unique factorization into NT-3 Prime Number Theorem IS-28 Principle extended pigeonhole EO-7 pigeonhole EO-5 Product of sets SF-2 Propositional logic BF-4, Lo-1 algebraic rules Lo-3 Public key cryptography NT-21 PGP NT-20 RSA protocol NT-23 Quantifier existential ( ) Lo-13 negation of Lo-15 universal ( ) Lo-12 Range of a function BF-1, SF-15 Rate of growth IS-18 Refinement of set partition SF-11, EO-16 Reflexive relation EO-3, EO-13 Relation antisymmetric EO-13 binary EO-3 covering EO-28 equivalence EO-1 functional inverse number of EO-15 order SF-8, EO-12 reflexive EO-3, EO-13 symmetric EO-3 transitive EO-3, EO-13 transitive closure of EO-26 Residue class (modular arithmetic) NT-6 Restriction of a poset (= subposet) EO-17 RSA protocol NT-23 Index-8
9 Rule absorption BF-6, Lo-3, SF-3 associative BF-6, Lo-3, SF-3 bound BF-6, Lo-3 commutative BF-6, Lo-3, SF-3 DeMorgan s BF-6, Lo-3, SF-3 distributive BF-6, Lo-3, SF-3 double negation BF-6, Lo-3, SF-3 idempotent BF-6, Lo-3, SF-3 negation BF-6, Lo-3 Sequence IS-12 algebraic rules for IS-16 bounded IS-16 convergent IS-13 convergent to infinity IS-19 decreasing IS-17 divergent IS-13 divergent to infinity IS-19 increasing IS-17 limit of IS-13 monotone IS-17 series and IS-20 tail of IS-12 term of IS-12 Series IS-20 Abel s Theorem IS-28 absolute convergence IS-26 alternating IS-24 alternating harmonic IS-23 conditional convergence IS-27 convergent IS-20 convergent and small terms IS-21 Dirichlet s Theorem IS-24 divergent IS-21 general harmonic IS-25 geometric IS-22 harmonic IS-22 integral test for monotone IS-24 partial sums IS-20 sum is a limit IS-20 tail of IS-20 Set SF-1 algebraic method SF-5 algebraic rules SF-2 as a predicate Lo-14 Bell number: B n SF-11 cardinality of SF-1 Cartesian product SF-2 characteristic function SF-10 complement SF-2 countable NT-5 difference SF-2 element method SF-4 empty SF-2 intersection SF-2 number of subsets SF-11 ordered SF-1 partially ordered EO-13 power SF-9, EO-13 subset SF-1 symmetric difference SF-2 union SF-2 universal SF-1 Venn diagrams SF-3 Set inclusion order EO-13 Set partition SF-11 block of SF-11 function coimage and SF-23 refinement poset EO-16 refining SF-11 Stirling number: S(n, k) SF-24 Sort bucket EO-22 comparison EO-22 topological (= linear extension) EO-30 Statement form Lo-1 Boolean function and Lo-8 Statement variable BF-3, Lo-1 Stirling number S(n,k) SF-24 String (= ordered set) SF-1 Subposet EO-17 Subset lattice EO-13 Subset of a set SF-1 number of them SF-11 Subset sums EO-6 Index-9
10 Sufficient (logic) Lo-7 Sum of powers IS-5 Sums equal EO-6 equal subset EO-6 Surjective function SF-18 Symmetric difference of sets SF-2 Symmetric encryption NT-20 Symmetric relation EO-3 Tabular form of a Boolean function BF-1 Tail and convergence IS-13 sequence IS-12 series IS-20 Tautology Lo-2 Term of a sequence IS-12 series IS-20 Theorem Abel s IS-28 algebraic rules, see Algebraic rules Pigeonhole Principle EO-5 Pigeonhole Principle, extended EO-7 Prime Number IS-28 sequence convergence, see Convergence Unique Factorization NT-3 There exists (logic: ) Tiling problem EO-24 Topological sort EO-30 Lo-13 Total order (= linear order) EO-14 Transitive closure EO-26 Transitive relation EO-3, EO-13 Trapdoor function NT-21 discrete logarithm NT-22 Truth set (predicate logic) Lo-12 Truth table BF-2, BF-4, Lo-2 Twin Prime conjecture Lo-16 Two-line notation SF-20, SF-20 Two s complement BF-16 arithmetic BF-16 overflow BF-17 Unary operator BF-3 Union of sets SF-2 Unique prime factorization NT-3 Universal quantifier ( ) Lo-12 Universal set SF-1 Unless (logic) Lo-8 Vector (= ordered set) SF-1 Venn diagrams for sets SF-3 Word (= ordered set) SF-1 Index-10
586 Index. vertex, 369 disjoint, 236 pairwise, 272, 395 disjoint sets, 236 disjunction, 33, 36 distributive laws
Index absolute value, 135 141 additive identity, 254 additive inverse, 254 aleph, 465 algebra of sets, 245, 278 antisymmetric relation, 387 arcsine function, 349 arithmetic sequence, 208 arrow diagram,
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